Integrable generalizations of oscillator and Coulomb systems via action-angle variables

Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N−1)-dimensional system. We present the action-angle formulation of such models in terms of the radial degree of freedom and the action-angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.     

The Poincaré-Lyapounov-Nekhoroshev Theorem

We give a detailed and mainly geometric proof of a theorem by N. N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1≤k≤n. This state's persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov theorem (corresponding to k=1) and the Liouville-Arnold one (corresponding to k=n) and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.     

Effects of finite size and shape on scattering and thermodynamic quantities at 2D Ising critical points

Using recent results by Cardy based on the conformal invariance of critical correlation functions we calculate universal results for scattering functions S(k), susceptibilities, correlation lengths and specific heat correction terms for finite Ising systems in two dimensions with circular and rectangular shapes and free boundary conditions. Our results specify the effect of shape on these quantities at the critical point. In particular, the half-width and lineshape of the scattering function is found to be strongly influenced by geometry. For a circle, S(k) follows the infinite system behavior 1/k^2−η, η = 0.25 only for very large k. For a substantial range of intermediate k values it is well represented by 1/k^2−η_app, with an “apparent” exponent η_app. We also discuss the probable influence of end, edge and domain wall effects on the correlation lengths, susceptibilities and specific heat correction terms. The application of our results to experimental systems and other theoretical models is considered.     

Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence

The nonlinear realization of conformal so(2,d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS_d+1 of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS₂ × S^d−1. The list of nonrelativistic systems “unified” by such a correspondence comprises the conformal mechanics model, the planar charge-vortex and three-dimensional charge-monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner-Nordström black hole.     

On the configuration of systems of interacting particles with minimum potential energy per particle

Previous work on infinite one-dimensional systems of interacting particles is continued. In the case of two-body potentials φ(x) = φ(-x), whose Fourier transform ?f(k) eicsts, it is shown that a necessary condition that the equidistant configuration has for a certain range of densities minimum potential energy per particle among all configurations of the same density, is that ?f(k)?0 for all k. An analogous theorem is proved for systems of particles in two and three dimensions.Furthermore some properties are discussed of one-dimensional systems for which ?f(k)?0 for all k and moreover ?f(k) = 0 for |k|?k₀.     

Uncovering Fractional Monodromy

The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n ₁:(?n ₂) resonant Hamiltonian systems with n ₁, n ₂ coprime natural numbers. We consider, in particular, systems that for n ₁, n ₂ > 1 contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n ₁, n ₂. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.     

Optimal oscillation-center transformations

Systems with Hamiltonians of the form H₀(p) + H₁(q,p,t) are considered. A variational principle is proposed for defining that canonical transformation, continuously connected with the identity transformation, which minimizes the residual, coordinate-dependent part of the new Hamiltonian. The principle is based on minimization of the mean square generalized force over phase space and time. The transformation reduces to the action-angle transformation in that part of the phase space of an integrable system where the orbit topology is that of the unperturbed system, or on primary KAM surfaces. General arguments in favour of this definition are given, based on Galilean invariance, decay of the Fourier spectrum, and its ability to include external fields or inhomogeneous systems. The optimal oscillation-center transformation for the physical pendulum (or particle in a sinusoidal potential) is constructed analytically. A modified principle for relativistic systems is presented in an appendix.     

Lie algebra contractions on two-dimensional hyperboloid

The Inönü-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E ₂ and eight on E _1,1. The text was submitted by the authors in English.     

Systems with separable many-particle interactions. I

In this paper we give an exact evaluation of the free energy per particle for systems with separable many-particle interactions described by a hamiltonian of the type ? = ∑_kT(k) + NP (N^-1 ∑_kV(k)), where P is an arbitrary polynomial. In the proof use is made of a fundamental theorem due to Bogoliubov Jr. for ferromagnetic quadratic operators. The free energy can be obtained from a trial hamiltonian, which is linear in the operators T and V.     

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL _k Heisenberg chain is dual to the special reduced k + 2-points gl _N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.     

Self-consistent calculation of the optical potential and ground-state properties of nuclear matter

On the basis of the Green-function formalism, we performed a self-consistent calculation of the self-energy ∑(k, ω) of a particle interacting with the infinite nuclear medium. The function ∑(k, ω) was mapped out in the energy-momentum plane, and the single-particle energy ω(k), momentum distribution ?(k) and the “on-shell” part of the self-energy, ∑(k, ω(k)), were defined, from which all physical properties followed. In particular we investigated the ground-state properties of nuclear matter in two Λ-approximations of the T-matrix. In one, the intermediate two-particle propagator, Λ₀₀, represented free-particle propagation; in the other, called Λ₁₁, intermediate states included both interacting particles and holes. Pauli principle effects were included in both approximations. The second approximation was expected to be conserving because it included a large part of the rearrangement effects which, we found, contributed ～6 MeV per particle to the average energy and ～28 MeV to the singleparticle energy at zero momentum. The Hugenholtz-van Hove theorem was nearly satisfied, with only 1 MeV separating the chemical potential from the average energy. We also studied, in the Λ₀₀-approximation, the optical potential for the scattering of a particle by a large nucleus; it was directly related to the “on-shell” part of the self-energy. It was found that, below 100 MeV, the real part varied as (?90 + 0.584E) [MeV], and the imaginary part as (2.4 + 0.009 E) [MeV].     

Thermal correction to resistivity in dilute Si-MOSFET two-dimensional systems

Neglecting electron-electron interactions and quantum interference effects, we calculate the classical resistivity of a two-dimensional electron (hole) gas, taking into account the degeneracy and the thermal correction due to the combined Peltier and Seebeck effects. The resistivity is found to be a universal function of the temperature, expressed in units of (h/e²)(k_Fl)^?1. Analysis of the compressibility and thermopower points to the thermodynamic nature of the metal-insulator transition in two-dimensional systems. We reproduce the beating pattern of Shubnikov-de Haas oscillations in both the crossed field configuration and Si-MOSFET valley splitting cases. The consequences of the integer quantum Hall effect in a dilute Si-MOSFET two-dimensional electron gas are discussed. The giant parallel magnetoresistivity is argued to result from the magnetic-field-driven disorder.     

Quantification of surface area and intrinsic mass transfer coefficient for ultrasound-assisted dissolution process of a sparingly soluble solid dispersed in aqueous solutions

The efficacy of power ultrasound of 20 kHz in enhancing the volumetric mass transfer coefficient was investigated in this study. Breakage and dissolution of sparingly soluble benzoic acid dispersed in either water or 24% aqueous glycerol was monitored as a function of time and ultrasound power input. Particle size measurements were carried out at intermediate times during the experiment to estimate the mean particle size and surface area. Linear combination of lognormal distributions was found to fit the experimental particle size distribution data. The De Brouckere mean diameters (d₄₃) obtained from the particle size distributions decreased with increase in the ultrasonic power level. Empirical correlations were developed for the evolution of surface area as a function of ultrasonic energy input per unit mass. The effect of ultrasound on the intrinsic mass transfer coefficient (k_c) could be decoupled from the volumetric mass transfer coefficient (k_ca) as the surface area was also estimated. Different approaches involving either constant or variable intrinsic mass transfer coefficients were employed when carrying out the delineation. Mass transfer rates were enhanced due to both higher ultrasound induced intrinsic convective mass transfer coefficient and additional surface area created from particle breakage. To delineate the effects of particle breakage from solid dissolution, experiments were also carried out under non-mass transfer conditions by pre-saturating the solvents with benzoic acid. Both the solid-liquid systems examined in the present study attained saturation concentration when the ultrasonic energy input per unit mass was approximately 60 kJ/kg, irrespective of the ultrasonic power level setting.     

Scaling law for the fluid-solid phase transition in Yukawa systems (dusty plasmas)

Yukawa systems serve as models for plasmas and colloidal suspensions of charged particles. The state of these systems is determined by two dimensionless parameters: k = a/λ _D , which is the ratio of the mean interparticle distance to the Debye length λ _D , and Γ = Z _d ² e ²/aT _d , which is the ratio of the Coulomb potential energy to the particle temperature T _d (Z _d is the charge of each particle). We propose an empirical scaling law for the critical coupling parameter Γ _c needed for crystallization in Yukawa systems. The dependence of Γ _c on k is in good agreement with recent molecular dynamics simulations.     

Logarithmic correlation functions in two-dimensional turbulence

We consider the correlation functions of two-dimensional turbulence in the presence and absence of a three-dimensional perturbation, by means of conformal field theory. In the presence of three-dimensional perturbation, we show that in the strong coupling limit of a small scale random force, there is some logarithmic factor in the correlation functions of velocity stream functions. We show that the logarithmic conformal field theory c_8,1 describes the 2D-turbulence both in the absence and in the presence of the perturbation. We obtain the energy spectrum E(k) ∼ k^−5.125 ln(k) for perturbed 2D-turbulence and E(k) ∼ k⁻⁵ ln(k) for unperturbed turbulence. Recent numerical simulation and experimental results confirm our prediction.     

Spin-wave interaction in two-dimensional ferromagnets with dipolar forces

We discuss the spin-wave interaction in two-dimensional (2D) Heisenberg ferromagnet (FM) with dipolar forces at T_C?T?0 using 1/S expansion. A comprehensive analysis is carried out of the first 1/S corrections to the spin-wave spectrum. In particular, we obtain that the spin-wave interaction leads to the gap in the spectrum ε_k renormalizing greatly the bare gapless spectrum at small momenta k. Expressions for the spin-wave damping Γ_k are derived self-consistently and it is concluded that magnons are well-defined quasi-particles in both quantum and classical 2D FMs at small T. We observe thermal enhancement of both Γ_k and Γ_k/ε_k at small momenta. In particular, a peak appears in Γ_k and Γ_k/ε_k at small k and at any given direction of k. If S∼1 the height of the peak in Γ_k/ε_k is not larger than a value proportional to T/D?1, where D is the spin-wave stiffness. In the case of large spins S?1 the peak in Γ_k/ε_k cannot be greater than that of the classical 2D FM found at k=0 whose height is small only numerically: Γ₀/ε₀≈0.16 for the simple square lattice.     

Wk structure of A SU(2)-level k from covariant construction of Kac-Moody algebras

The W_k structure underlying the transverse realization of affine SU(2) at level k is analyzed. The extension of the equivalence existing between the covariant and light-cone gauge realization of an affine Kac-Moody algebra to W_k algebras is given. Higher spin generators are extracted by the less singular terms in the operator product expansion of the parafermions constructed by means of the projection of the covariant on the light-cone gauge. These fields can be written in terms of only one free boson compactified on a circle.     

On phase transitions in Schlögl's second model

We study Schlögl's second model, characterized by chemical reactions $$\begin{array}{*{20}c} {2X\underset{{k_2 }}{\overset{{k_1 }}{\longleftrightarrow}}3X,} & {X\underset{{k_4 }}{\overset{{k_3 }}{\longleftrightarrow}}0,} \\ \end{array} $$ ind-dimensional space. The reactions are assumed to be local; local fluctuations are fully taken into account, and particle transport occurs via diffusion. In contrast to previous investigations, we find no phase transition whenk ₄≠0 andd<4. Fork ₄=0,k ₃≠0, and 1≦d<4, we find a second-order phase transition which is in the same universality class as the transition in Schlögl's first model. Only ford≧4 we do find the first-order transition found also by previous authors. These claims are supported by extensive Monte Carlo calculations for various realizations of this process on discrete space-time lattices.     

Separation of cross-relaxation and exchange in two-site spin systems without resolved couplings

Variable temperature two-dimensional nuclear Overhauser enhancement experiment (2-D NOESY) is used to extract the rate constants and cross-relaxation rates that contribute to the same cross-peaks in NOESY spectra. Rate constants (k _AB) and cross-relaxation rates (R _AB) for two-site spin systems are related to the ratio between the cross-peak and diagonal peak integrals (F) by the expression:R _AB -k _AB = (1/2τ _m)ln[(1 -F)/(1 +F)], where τ_m is the mixing time. As a model, we investigated the exchange processes in a system of dimer calix[4]arenes of C_4v symmetrical configuration with guest inclusion (benzene or benzene-d₆), where the measurement of exchange processes is hindered by the presence of strong nuclear Overhauser enhancement between protons in adjacent aromatic rings in the cone conformation of the calix[4]arene.     

Adsorbate band dispersions for C on Ru(0001)

We have studied carbon-induced two-dimensional energy bands on Ru(0001) using angleresolved photoelectron spectroscopy and have compared them with ab initio calculations. We find a nearly parabolic band (bottom at E_F ?9.8 eV at k_∥ = 0, effective mass ～ 1.5 m_e) which we assign to the C 2p_z valence states of a graphitic carbon overlayer. Compared to graphite, these states are bound more tightly by 2.3 eV.